1. Plasma Dynamos UCLA January 5th 2009 Steve Cowley, UKAEA Culham and Imperial Thanks to Alex Schekochihin, Russell Kulsrud, Greg Hammett and Mark Rosin.

2. Early magnetic fields -- what, when and how. After re-ionization the universe was probably a reasonably collisionless turbulent high  plasma. Many large scale plasmas are quite collisionless. I will argue that (random) magnetic fields grow rapidly in such a plasma. I will also argue that we need to know a lot more about the small scale dynamics of high  plasmas. We need an experiment at  >> 1!

3. Cluster Turbulence The Coma Cluster: pressure map [Schuecker et al. 2004, A&A 426, 387] L ~ 10 2 …10 3 kpc U ~ 10 2 …10 3 km/s (subsonic) L/U ~ 10 8 …10 9 yr Mergers AGNs Wakes

4. Cluster Turbulence Note: it is not obvious that there is turbulence! [A. Fabian 2003, MNRAS 344, L48] Mergers AGNs Wakes L ~ 10 2 …10 3 kpc U ~ 10 2 …10 3 km/s (subsonic) L/U ~ 10 8 …10 9 yr mfp ~ 0.1…10 kpc Re ~ 1…10 2 The Coma Cluster [Schuecker et al. 2004, A&A 426, 387]

5. Cluster Magnetic Fields Abell 400 cluster [Eilek & Owen 2002, ApJ 567, 202] 900 kpc

6. Cluster MHD Turbulence Turbulence scale is around here TURBULENCE Coma cluster [Schuecker et al. 2004, A&A 426, 387] MAGNETIC FIELDS Hydra A Cluster [Vogt & Enßlin 2005, A&A 434, 67] Magnetic Reynolds #, Rm ~ 10 29.Magnetic Reynolds #, Rm ~ 10 29.

7. The Large Prandtl Number Case: Galaxies, Clusters etc. Magnetic Prandtl number = Pr = / . On the turnover time of the viscous eddies the “seed field” grows. The field develops structure below the viscous scale down to the resistive scale l  = Pr -1/2 l l l = 10 -30kpc Viscous scale t = 10 8 years Viscous eddy Turnover.

8. Isotropic Homogeneous Dynamo Folded Structure at Resistive Scale Grayscale is |B|. Scalar Viscosity

9. Plasma not Fluid

10. Magnetized Viscosity --Anisotropic Pressure Anisotropic pressure tensor in magnetized plasma. Because of fast motion around the field the tensor must be of the form: DEFINITION OF PRESSURE TENSOR.

11. Magnetized Viscosity. B Collisionless particle motion restricted to being close to field line and conserving . Compressing Field Collisionless. Relaxed by Collisions. P

12. Incompressible Braginskii MHD. Coefficients worked out by Braginskii Reviews of Plasma Physics Vol. 2. Collisional limit Unit vector along B

13. Equilibrium -- Decreasing B. V0V0 V0V0 B0B0 B0B0 Stretching rate

14. Firehose Instability. Look at instabilities that are smaller scale than the field and growing faster than the stretching rate. Treat B 0 as quasi-constant during the growth. We take perturbed velocity to go as: The condition that the growth rate is faster than stretching rate is:

15. Firehose Instability. Linearized: The x component becomes: Perturbed field line Curvature.

16. Firehose Instability. Putting this into force equation we get. Alfven wave when no anisotropy

17. More Firehose. Parallel pressure forces squeeze tube out. Rosenbluth 1956 Southwood and Kivelson 1993 P || Tighter bend grows faster. Unstable when Growth rate at negligible B

18. So What!? -- Nonlinear Firehose. Nonlinear kinetic theory gives: Rate of change of B 2 averaged along B. Instability tries to keep average B constant by bending the field. Diverging Flow. Makes finite wiggles Schekochihin et. al. Phys. Rev. Lett.

19. Nonlinear Mirror Mode. When the field increasing the plasma is unstable to the mirror Mode which creates little traps in the plasma. Converging

20. Stretching and compressing Field decreasing P || >P  Firehose Unstable. Field increasing P || < P  Mirror mode Unstable. Stretched at the turnover rate of the viscous eddies. Using Braginskii’s Expression we get P || -P  ~ Re -1/2 P ~ P/6

21. Scales l 0 ~ 1-3Mpc Viscous Scale l ~ l 0 Re -3/4 ~ 10 - 30 kpc l 0 /u 0 ~10 9 years l / u ~10 8 years Mean-Free Path. mfp ~ l 0 Re -1 ~1-10kpc k Resistive Scale ~ l 0 Rm -1/2 ~ 10 4 km Ion Larmor Radius Scale @ B = 1  G ~10 5 km. E B ? EVEV   Maximum growth rate

22. l 0 ~ 1-3Mpc Viscous Scale l ~ l 0 Re -3/4 ~ 10 - 30 kpc l 0 /u 0 ~10 9 years l / u ~10 8 years Mean-Free Path. mfp ~ l 0 Re -1 ~1-10kpc k Resistive Scale ~ l 0 Rm -1/2 ~ 10 4 km Ion Larmor Radius Scale @ B = 1  G ~10 5 km. E B ? EVEV ? ? Scales

23. What does small scale field do? Enhanced particle scattering? Effective collisions increase -- i ? If so viscosity decreases -- Re gets large and turbulence has faster motions. Dynamo Growth Time:  ~  0 (  i /L) (1/2) ~ 1000 years! MAGNETIC FIELD CAN GROW ON TRIVIAL TIMESCALES. Sharma, Hammett, Schekochihin, Kulsrud etc.

24. Conclusions. Small scale fast growing instabilities to be expected in weak field magnetized fully ionized plasmas. Make finite wiggles on the scale almost of the ion larmor radius. May enhance collisions, dissipation and change the transport properties.